at the level of the corresponding \(\infty \) -categories, thus giving a classical description of the left side in the case of rigid spaces, see [1]}. In fact, we could have defined \(D_{\mathop {\mathrm {\acute{e}t}}}(X,\mathbf {Z}_\ell )\)  as the homotopy category of the right side above, and thus avoided ever mentioning the ambient category \(D_{\ell -\text{comp}}(X_v,\mathbf {Z}_\ell )\) ; one reason we introduce the latter is that it carries an obvious \(t\) -structure, which we shall use in our proofs.
